How do these geometry figures affect riding?
 

How do front center, chain stay length, head tube angle affect riding?

Front center and chainstay length set your fore-aft weight distribution. If the proportions are right, you get better handling and possibly stability.

Head tube angle affects the tendency of the fork to flop (called flop), and combined with fork offset and wheel diameter determines trail, which is really important in designing how your frame will steer.

FC and CS have something to do with fit, and head angle less so. But, a more laid-back head tube can result in longer FC, so it's not like there's no connection to fit.

Thanks, Road Fan.

A short Front Centre creates toe-clip overlap, which is undesirable for low-speed urban riding or loaded riding.
Head-tube angle affects the sensitivity of the bike to changes in fork rake.
CS length affects rear-triangle siffness, pedal interference, as well as fore-aft weight distribution.

Would it be right to say that in general, shorter front center, greater seat (not head) tube angle, and shorter wheel base are more for speedy rides such as racing, and the opposite are more for leisure/urban/touring/slow riding?